Question

An isolated system has two phases, denoted by A and B, each of which consists of the same two substances, denoted by 1 and 2. The phases are separated by a freely moving, thin wall permeable only by substance 2 . Determine the necessary conditions for equilibrium.

Short answer

For equilibrium, chemical potential, temperature, and pressure must be equal in both phases: \( \mu_2^A = \mu_2^B \), \(T^A = T^B\), and \(P^A = P^B\).

Step-by-step solution

Understand the System

Recognize that the system consists of two phases, A and B, separated by a thin wall permeable only by substance 2. Both phases contain substances 1 and 2.

Apply the Chemical Potential Equilibrium Condition

For equilibrium, the chemical potential of substance 2 must be equal in both phases A and B. This is given by: \( \mu_2^A = \mu_2^B \) where \( \mu_2^A \) and \( \mu_2^B \) are the chemical potentials of substance 2 in phases A and B, respectively.

Apply the Phase Rule

Since the phases are in equilibrium and the wall is permeable only to substance 2, we need to account for the fact that substance 2 can pass freely between the two phases. The number of components (C) is 2, and the number of phases (P) is 2. The degrees of freedom (F), given by the formula \( F = C - P + 2 \), equals 2.

Express General Condition for Mechanical and Thermal Equilibrium

For mechanical and thermal equilibrium, the temperature and pressure must also be the same in both phases. Thus, \(T^A = T^B\) and \(P^A = P^B\).

Combine Conditions

Combine the conditions of chemical, thermal, and mechanical equilibrium to find that for the system in equilibrium, the following must hold: \( \mu_2^A = \mu_2^B \), \(T^A = T^B\), and \(P^A = P^B\).

Key concepts

Chemical Potential

Chemical potential is a key concept in understanding equilibrium in isolated systems, especially when multiple phases are involved. It represents the change in the system's energy when the number of particles is changed. For any substance in a phase, the chemical potential is denoted by \( \mu_i \), where ``i`` is the specific substance.

If a system is at equilibrium, the chemical potential of a substance must be the same in all phases where it exists. In the problem you looked at, we have two phases called A and B. These phases are separated by a thin wall that is permeable only to substance 2. For equilibrium in substance 2, this means:

\( \mu_2^A = \mu_2^B \)

No flow of substance 2 will occur between the phases as long as this condition is met. This is because the system is stable, and the energy will not change with the movement of substance 2 between the phases.

This concept is critical in many chemical and physical contexts, such as separating mixtures, understanding battery functions, and even biological systems where membranes play a crucial role.

Degrees of Freedom in Thermodynamics

Degrees of freedom (F) in thermodynamics refer to the number of independent variables that can change without affecting the others. This is a vital concept in describing the state of a system and predicting its behavior during a phase change or under different conditions.

In the given exercise, to determine the degrees of freedom, we use the Gibbs Phase Rule:

\( F = C - P + 2 \)

Here, ``C`` is the number of components and ``P`` is the number of phases. For this system:

- Number of components (C): 2
- Number of phases (P): 2

Plugging these values into the formula, we get:

\( F = 2 - 2 + 2 = 2 \)

This implies there are two degrees of freedom, meaning we can change two variables (like temperature and pressure) independently without affecting the equilibrium condition.

This calculation is essential for understanding how a system behaves when subjected to changes and helps us predict how the system can be manipulated to reach a desired state.

Phase Equilibrium

Phase equilibrium occurs when two or more phases (solid, liquid, gas) of a substance coexist without any net change over time. This state implies that the fundamental properties such as temperature, pressure, and chemical potential are balanced across the phases.

In the exercise, phases A and B are separated by a thin wall permeable only to substance 2. When considering phase equilibrium, we need to meet certain conditions:

- Chemical Equilibrium: Ensure the chemical potential of substance 2 is equal in both phases:
\( \mu_2^A = \mu_2^B \)

- Thermal Equilibrium: The temperatures in both phases must be equal:
\( T^A = T^B \)

- Mechanical Equilibrium: The pressures in both phases must be equal:
\( P^A = P^B \)

When all these conditions are satisfied, we have reached phase equilibrium. This balanced state means no net energy or mass transfer occurs between the phases, stabilizing the system.

Understanding phase equilibrium is vital in various scientific and industrial systems, including material science, chemical engineering, and thermodynamics. It helps in designing processes like distillation, crystallization, and even in predicting the weather patterns by understanding phase changes in the atmosphere.